The penitent has no form that he brings to Confession; that’s what the Priest does.
If by “faithful Catholics” you mean “penitents,” then you again are confusing form and matter. Confessed sins are the matter of Confession. They are not the form.
Like all Sacraments, the proper form and matter are required.
It is a genus of sin, correct. However, where you’re confusing yourself is that you’re presupposing that a particular word is not being understood to refer to a particular action because it is a genus. Words have a semantic range. In one context, a word could mean one thing: such as “unchastity” being a genus of sin. But in another context, it can mean something different: such as when a Priest asks you to use that word when you mean a particular action.
The Priest is not confused as to what you’re talking about. He knows what you mean. You know that he knows what you mean. But your fear is that you are not technically fulfilling a duty while ignoring that in practice you did. That is rooted in legalism.
The matter of Confession is the confessed sin which is itself rooted in the intent of the penitent. This is absolutely different from any of the other Sacraments, whose matter does not depend on intent, but rather depends on specific externals.
There isn’t a library of old books stashed away in the Vatican that describes precisely the exact wording that must be used for a particular sin in order for it to properly constitute the matter of Confession. There is no such external and acting as if there is is legalistic.
But rather, there is the requirement for mortal sins to be named by species and by number … so that the Priest understands what the species and number are. And HOW they are named by species can absolutely be influenced by the direction of the Confessor. He won’t simply tell you to say X when you mean Y, and somehow your saying X in fact implies Z.
I can mean to say something in Confession when I use certain words that might not accurately describe what I’m referring to. And that intention suffices to be the valid matter of my Confession. Therein, my intention is important when it comes to matter.
But again, in other Sacraments, where the matter is precisely specified, intention cannot simply transform invalid matter. Therein, intention is not important when it comes to matter.